In devices for reproducing information recorded on recording media such as a HDD, a DVD, and a Blu-ray Disc, PRML signal processing techniques have been employed to improve playback performance. As for PR equalization, adaptive equalization methods have been employed to respond to variations in characteristics of recording media and playback paths.
Currently, there is generally employed an adaptive equalization method that minimizes an equalization error by a LMS (Least Mean Square) algorithm, while another algorithm that minimizes an error in maximum likelihood decoding is also proposed for further improvement in playback performance. For example, there is proposed an adaptive equalization method which updates coefficients so as to minimize an error in maximum likelihood decoding, which error is obtained from SAM (Sequenced Amplitude Margin) (refer to “Tetsuya Okumura et al., New Adaptive Equalization Method for PRML System using Sequenced Amplitude Margin, Technical Digest of Optical Data Storage 2003, May 11th, 2003, pp.96-98”). Hereinafter, a description will be given of coefficient update formulas obtained when the aforementioned method is applied to (1,7)RLL (Run-length Limited coding), PR(1,2,2,1)ML, with reference to FIGS. 7 and 8. FIG. 7 shows an equalization waveform of ideal PR(1,2,2,1). The ideal PR(1,2,2,1) equalization waveform in (1,7)RLL has seven amplitude values, which are represented as T0, T1, T2, T3, T4, T5, T6, respectively. FIG. 8 is a trellis diagram. The state of the trellis diagram is determined by 3 bits, and it is denoted as S000 when the bit sequence is (000).
The ideal waveform for a given bit sequence (0001111) is (T1, T3, T5, T6) that is shown in FIG. 7 by the solid line, and the state of its trellis diagram changes from S000 to S111 as shown in FIG. 8 by the solid line. Likewise, the ideal waveform for a given bit sequence (0000111) is (T0, T1, T3, T5) that is shown in FIG. 7 by the dotted line, and the state of its trellis diagram changes from S111 to S000 as shown in FIG. 8 by the dotted line.
To be specific, there exist two paths in the same state transition, and the square of Euclidean distance d2 between these two paths can be calculated as follows.d2=(T1−T0)2+(T3−T1)2+(T5−T3)2+(T6−T5)2   (1)
The following table 1 provides a summary of the square of Euclidean distance d2 between two paths having the same state transition, in bit sequences each having a length of 7 bits.
TABLE 1State Transition and Square of Euclidean Distance in(1, 7)RLL + PR(1, 2, 2, 1)StateTransitionSequence 1Sequence 2d2S000→S11100011110000111(T1 − T0)2 + (T3 − T1)2 +(T5 − T3)2 + (T6 − T5)2S000→S11000011100000110(T1 − T0)2 + (T3 − T1)2 +(T5 − T3)2 + (T5 − T4)2S100→S11110011111000111(T2 − T1)2 + (T3 − T1)2 +(T5 − T3)2 + (T6 − T5)2S100→S11010011101000110(T2 − T1)2 + (T3 − T1)2 +(T5 − T3)2 + (T5 − T4)2S011→S00101110010110001(T5 − T4)2 + (T5 − T3)2 +(T3 − T1)2 + (T2 − T1)2S011→S00001110000110000(T5 − T4)2 + (T5 − T3)2 +(T3 − T1)2 + (T1 − T0)2S111→S00111110011110001(T6 − T5)2 + (T5 − T3)2 +(T3 − T1)2 + (T2 − T1)2S111→S00011110001110000(T6 − T5)2 + (T5 − T3)2 +(T3 − T1)2 + (T1 − T0)2
If the seven values are all equally spaced, that is, if the seven values can be expressed by the following equation,(T1−T0)=(T2−T1)=(T3−T2)=(T4−T3)=(T5−T4)=(T6−T5)   (2)the squares d2 of the 8 state transitions would have the same value, 10(T1−T0)2.
It is now assumed that a waveform corresponding to a correct bit sequence (0001111) is input to an equalizer. The bit sequence corresponding to another path having the same state transition is (0000111), which is the most frequently mis-decoded bit sequence during maximum likelihood decoding, and therefore, it is called as an incorrect bit sequence.
Assuming that the input waveform is {u(−4,n), u(−3,n), u(−2,n), u(−1,n), u(0,n), u(1,n)}, the equalization waveform is {y(−3,n), y(−2,n), y(−1,n), y(0,n)}, and the tap coefficient is c(k,n), the following equations will hold.
                                                                                                                                                                                                                  y                            ⁡                                                          (                                                              0                                ,                                n                                                            )                                                                                =                                                                                    ∑                                                              k                                =                                                                  -                                  1                                                                                            1                                                        ⁢                                                                                          c                                ⁡                                                                  (                                                                      k                                    ,                                    n                                                                    )                                                                                            ⁢                                                              u                                ⁡                                                                  (                                                                      k                                    ,                                    n                                                                    )                                                                                                                                                                                                                                                                                                              y                            ⁡                                                          (                                                                                                -                                  1                                                                ,                                n                                                            )                                                                                =                                                                                    ∑                                                              k                                =                                                                  -                                  1                                                                                            1                                                        ⁢                                                                                          c                                ⁡                                                                  (                                                                      k                                    ,                                    n                                                                    )                                                                                            ⁢                                                              u                                ⁡                                                                  (                                                                                                                                                    -                                        1                                                                            +                                      k                                                                        ,                                    n                                                                    )                                                                                                                                                                                                                                                                                                                                y                      ⁡                                              (                                                                              -                            2                                                    ,                          n                                                )                                                              =                                                                  ∑                                                  k                          =                                                      -                            1                                                                          1                                            ⁢                                                                        c                          ⁡                                                      (                                                          k                              ,                              n                                                        )                                                                          ⁢                                                  u                          ⁡                                                      (                                                                                                                            -                                  2                                                                +                                k                                                            ,                              n                                                        )                                                                                                                                                                                                                                      y                ⁡                                  (                                                            -                      3                                        ,                    n                                    )                                            =                                                ∑                                      k                    =                                          -                      1                                                        1                                ⁢                                                      c                    ⁡                                          (                                              k                        ,                        n                                            )                                                        ⁢                                      u                    ⁡                                          (                                                                                                    -                            3                                                    +                          k                                                ,                        n                                            )                                                                                                                              (        3        )            
The above equations represent a 3-tap FIR (Finite Impulse Response) type equalizer, which can be illustrated as FIG. 9. While in FIG. 9 the number of taps is three for convenience sake, an expansion of the number of taps can easily be achieved.
The ideal PR(1,2,2,1) equalization waveforms corresponding to the correct bit sequence (0001111) and the incorrect bit sequence (0000111) are (T1, T3, T5, T6), (T0, T1, T3, T5), respectively. Consequently, the metric difference s(n) between the equalization waveform and each of the two paths can be calculated as follows:
                                                                        s                ⁢                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        0                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                                            {                                                                        y                          ⁡                                                      (                                                                                          -                                3                                                            ,                              n                                                        )                                                                          -                                                  T                          1                                                                    }                                        2                                    ⁢                                                            {                                                                        y                          ⁡                                                      (                                                                                          -                                2                                                            ,                              n                                                        )                                                                          -                                                  T                          3                                                                    }                                        2                                                  -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        6                                                              }                                    2                                                                                                        =                            ⁢                                                2                  ⁢                                      {                                                                  T                        1                                            -                                              T                        0                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                  y                  ⁢                                      (                                                                  -                        2                                            ,                      n                                        )                                                  +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        6                                            -                                              T                        5                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  T                  0                  2                                -                                  T                  6                  2                                                                                                        =                            ⁢                              2                ⁢                                                      ∑                                          k                      =                                              -                        1                                                              1                                    ⁢                                                            c                      ⁡                                              (                                                  k                          ,                          n                                                )                                                              ⁢                                          {                                                                                                    {                                                                                          T                                1                                                            -                                                              T                                0                                                                                      }                                                    ⁢                                                      u                            ⁡                                                          (                                                                                                                                    -                                    3                                                                    +                                  k                                                                ,                                n                                                            )                                                                                                      +                                                                              {                                                                                          T                                3                                                            -                                                              T                                1                                                                                      }                                                    ⁢                          u                          ⁢                                                      (                                                                                                                            -                                  2                                                                +                                k                                                            ,                              n                                                        )                                                                          +                                                                                                                                                                                                    ⁢                                                                            {                                                                        T                          5                                                -                                                  T                          3                                                                    }                                        ⁢                                          u                      ⁡                                              (                                                                                                            -                              1                                                        +                            k                                                    ,                          n                                                )                                                                              +                                                            {                                                                        T                          6                                                -                                                  T                          5                                                                    }                                        ⁢                                          u                      ⁡                                              (                                                  k                          ,                          n                                                )                                                                                            }                            +                              T                0                2                            -                              T                6                2                                                                        (        4        )            
On the other hand, an error function ε is defined as follows:ε=E└{s(n)−d2}2┘  (5)where E represents the expected value operator. By substituting equations (4) into equation (5), and partial-differentiating ε with c(k,n), the following equation (6) can be obtained.
                                          ∂            ɛ                                ∂                          c              ⁡                              (                                  k                  ,                  n                                )                                                    =                  2          ⁢                      E            ⁡                          [                                                                    {                                                                  s                        ⁡                                                  (                          n                          )                                                                    -                                              d                        2                                                              }                                    ·                  2                                ⁢                                  {                                                                                    {                                                                              T                            1                                                    -                                                      T                            0                                                                          }                                            ⁢                                              u                        ⁡                                                  (                                                                                                                    -                                3                                                            +                              k                                                        ,                            n                                                    )                                                                                      +                                                                  {                                                                              T                            3                                                    -                                                      T                            1                                                                          }                                            ⁢                                              u                        ⁡                                                  (                                                                                                                    -                                2                                                            +                              k                                                        ,                            n                                                    )                                                                                      +                                                                  {                                                                              T                            5                                                    -                                                      T                            3                                                                          }                                            ⁢                                              u                        ⁡                                                  (                                                                                                                    -                                1                                                            +                              k                                                        ,                            n                                                    )                                                                                      +                                                                  {                                                                              T                            6                                                    -                                                      T                            5                                                                          }                                            ⁢                                              u                        ⁡                                                  (                                                      k                            ,                            n                                                    )                                                                                                      }                                            ]                                                          (        6        )            
The above equation (6) represents a gradient vector of an error characteristics curved-plane. The element of the gradient vector is equal to the primary derivative of the mean square error ε regarding the tap coefficient c(k,n). When the tap coefficient is continuously corrected in the opposite direction of the gradient vector, that is, in the steepest descent direction of the error characteristics curved-plane, the minimum mean square error εmin is reached eventually. This is a long-existing optimization method, called as the method of steepest descent.
In order to realize the above-described method as a practical device, the expected value operator E should be replaced by an instant estimation value. That is, the tap coefficient c(k,n) should be updated as below.c(k,n+1)=c(k,n)−2μ{s(n)−d2}·2{{T1−T0}u(−3+k,n)+{T3−T1}u(−2+k,n)+{T5−T3}u(−1+k,n)+{T6−T5}u(k,n)}  (7)where μ is the step size parameter. The above algorithm is a new adaptive equalization method.
While the above formula (7) is obtained from the correct bit sequence (0001111), similar formulas can also be obtained from other bit sequences. Hereinafter, coefficient update formulas for 16 correct bit sequences shown in table 1 will be described.
1) In the case of a correct bit sequence (0001111) and an incorrect bit sequence (0000111), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (8), (9), and (10), respectively.d207=(T1−T0)2+(T3−T1)2+(T5−T3)2+(T6−T5)2   (8)
                                                                                          s                  07                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        0                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        6                                                              }                                    2                                                                                                        =                            ⁢                                                2                  ⁢                                      {                                                                  T                        1                                            -                                              T                        0                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          2                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        6                                            -                                              T                        5                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  T                  0                  2                                -                                  T                  6                  2                                                                                        (        9        )                                                      ∂                                          s                07                            ⁡                              (                n                )                                                          ∂                          c              ⁡                              (                                  k                  ,                  n                                )                                                    =                              2            ⁢                          {                                                T                  1                                -                                  T                  0                                            }                        ⁢                          u              ⁡                              (                                                                            -                      3                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                                                            -                      2                                        +                    k                                    ,                  n                                )                                              +                    ⁢                      2            ⁢                          (                                                T                  5                                -                                  T                  3                                            }                        ⁢                          u              ⁡                              (                                                                            -                      1                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  6                                -                                  T                  5                                            }                        ⁢                          u              ⁡                              (                                  k                  ,                  n                                )                                                                        (        10        )            
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    07                                    ⁡                                      (                    n                    )                                                  -                                  d                  07                  2                                            }                        ⁢                                          ∂                                                      s                    07                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        11        )            
2) In the case of a correct bit sequence (0000111) and an incorrect bit sequence (0001111), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    07                                    ⁡                                      (                    n                    )                                                  -                                  d                  07                  2                                            }                        ⁢                                          ∂                                                      s                    07                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        12        )            
3) In the case of a correct bit sequence (0001110) and an incorrect bit sequence (0000110), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (13), (14), and (15), respectively.d206=(T1−T0)2+(T3−T1)2+(T5−T3)2+(T5−T4)2   (13)
                                                                                          s                  06                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        0                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        4                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                -                                                                                                      ⁢                                                                                          {                                                                        y                          ⁡                                                      (                                                                                          -                                1                                                            ,                              n                                                        )                                                                          -                                                  T                          5                                                                    }                                        2                                    -                                                            {                                                                        y                          ⁡                                                      (                                                          0                              ,                              n                                                        )                                                                          -                                                  T                          5                                                                    }                                        2                                                  =                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        1                                            -                                              T                        0                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          2                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  T                  0                  2                                -                                  T                  4                  2                                -                                  2                  ⁢                                      T                    5                    2                                                                                                          (        14        )                                                      ∂                                          s                06                            ⁡                              (                n                )                                                          ∂                          c              ⁡                              (                                  k                  ,                  n                                )                                                    =                              2            ⁢                          {                                                T                  1                                -                                  T                  0                                            }                        ⁢                          u              ⁡                              (                                                                            -                      3                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                                                            -                      2                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          (                                                T                  5                                -                                  T                  3                                            }                        ⁢                          u              ⁡                              (                                                                            -                      1                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  5                                -                                  T                  4                                            }                        ⁢                          u              ⁡                              (                                  k                  ,                  n                                )                                                                        (        15        )            
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    06                                    ⁡                                      (                    n                    )                                                  -                                  d                  06                  2                                            }                        ⁢                                          ∂                                                      s                    06                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        16        )            
4) In the case of a correct bit sequence (0000110) and an incorrect bit sequence (0001110), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    06                                    ⁡                                      (                    n                    )                                                  -                                  d                  06                  2                                            }                        ⁢                                          ∂                                                      s                    06                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        17        )            
5) In the case of a correct bit sequence (1001111) and an incorrect bit sequence (1000111), the square of Euclidean distance d2 between two paths having a same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (18), (19), and (29), respectively.d247=(T2−T1)2+(T3−T1)2+(T5−T3)2+(T6−T5)2   (18)
                                                                                          s                  47                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        2                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                -                                                                                                      ⁢                                                                                          {                                                                        y                          ⁡                                                      (                                                                                          -                                1                                                            ,                              n                                                        )                                                                          -                                                  T                          5                                                                    }                                        2                                    -                                                            {                                                                        y                          ⁡                                                      (                                                          0                              ,                              n                                                        )                                                                          -                                                  T                          6                                                                    }                                        2                                                  =                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        2                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          2                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        6                                            -                                              T                        5                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  2                  ⁢                                      T                    1                    2                                                  -                                  T                  2                  2                                -                                  T                  6                  2                                                                                        (        19        )                                                      ∂                                          s                47                            ⁡                              (                n                )                                                          ∂                          c              ⁡                              (                                  k                  ,                  n                                )                                                    =                              2            ⁢                          {                                                T                  2                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                                                            -                      3                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                                                            -                      2                                        +                    k                                    ,                  n                                )                                              +                    ⁢                      2            ⁢                          (                                                T                  5                                -                                  T                  3                                            }                        ⁢                          u              ⁡                              (                                                                            -                      1                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  6                                -                                  T                  5                                            }                        ⁢                          u              ⁡                              (                                  k                  ,                  n                                )                                                                        (        20        )            
Consequently, the tap coefficient c(k,n) is updated as the following.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    47                                    ⁡                                      (                    n                    )                                                  -                                  d                  47                  2                                            }                        ⁢                                          ∂                                                      s                    47                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        21        )            
6) In the case of a correct bit sequence (1000111) and an incorrect bit sequence (1001111), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                +                      2            ⁢            μ            ⁢                          {                                                -                                                            s                      47                                        ⁡                                          (                      n                      )                                                                      -                                  d                  47                  2                                            }                        ⁢                                          ∂                                                      s                    47                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        22        )            
7) In the case of a correct bit sequence (1001110) and an incorrect bit sequence (1000110), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (23), (24), and (25), respectively.d246=(T2−T1)2+(T3−T1)2+(T5−T3)2+(T5−T4)2   (23)
                                                                                          s                  46                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        4                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        2                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                -                                                                                                      ⁢                                                                                          {                                                                        y                          ⁡                                                      (                                                                                          -                                1                                                            ,                              n                                                        )                                                                          -                                                  T                          5                                                                    }                                        2                                    -                                                            {                                                                        y                          ⁡                                                      (                                                          0                              ,                              n                                                        )                                                                          -                                                  T                          5                                                                    }                                        2                                                  =                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        2                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          2                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  2                  ⁢                                      T                    1                    2                                                  +                                  T                  4                  2                                -                                  T                  2                  2                                -                                  2                  ⁢                                      T                    5                    2                                                                                                          (        24        )                                                                                                      ∂                                                            s                      46                                        ⁡                                          (                      n                      )                                                                                        ∂                                      c                    ⁡                                          (                                              k                        ,                        n                                            )                                                                                  =                            ⁢                                                2                  ⁢                                      {                                                                  T                        2                                            -                                              T                        1                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            3                                                    +                          k                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            2                                                    +                          k                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      (                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            1                                                    +                          k                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      u                    ⁡                                          (                                              k                        ,                        n                                            )                                                                                  ]                                                          (        25        )            
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    46                                    ⁡                                      (                    n                    )                                                  -                                  d                  46                  2                                            }                        ⁢                                          ∂                                                      s                    46                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        26        )            
8) In the case of a correct bit sequence (1000110) and an incorrect bit sequence (1001110), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                +                      2            ⁢            μ            ⁢                          {                                                -                                                            s                      46                                        ⁡                                          (                      n                      )                                                                      -                                  d                  46                  2                                            }                        ⁢                                          ∂                                                      s                    46                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        27        )            
9) In the case of a correct bit sequence (0111001) and an incorrect bit sequence (0110001), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (28), (29), and (30), respectively.d231=(T5−T4)2+(T5−T3)2+(T3−T1)2+(T2−T1)2=d246   (28)
                                                                                          s                  31                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        4                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                                                                      ⁢                                                                                          {                                                                        y                          ⁡                                                      (                                                                                          -                                1                                                            ,                              n                                                        )                                                                          -                                                  T                          3                                                                    }                                        2                                    -                                                            {                                                                        y                          ⁡                                                      (                                                          0                              ,                              n                                                        )                                                                          -                                                  T                          2                                                                    }                                        2                                                  =                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          2                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        2                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  T                  4                  2                                +                                  2                  ⁢                                      T                    1                    2                                                  -                                  2                  ⁢                                      T                    5                    2                                                  -                                  T                  2                  2                                                                                        (        29        )                                                                                                      ∂                                                            s                      31                                        ⁡                                          (                      n                      )                                                                                        ∂                                      c                    ⁡                                          (                                              k                        ,                        n                                            )                                                                                  =                            ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            3                                                    +                          k                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            2                                                    +                          k                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      (                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            1                                                    +                          k                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        2                                            -                                              T                        1                                                              }                                    ⁢                                      u                    ⁡                                          (                                              k                        ,                        n                                            )                                                                                  ]                                                          (        30        )            
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    31                                    ⁡                                      (                    n                    )                                                  -                                  d                  31                  2                                            }                        ⁢                                          ∂                                                      s                    31                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        31        )            
10) In the case of a correct bit sequence (0110001) and an incorrect bit sequence (0111001), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                +                      2            ⁢            μ            ⁢                          {                                                -                                                            s                      31                                        ⁡                                          (                      n                      )                                                                      -                                  d                  31                  2                                            }                        ⁢                                          ∂                                                      s                    31                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        32        )            
11) In the case of a correct bit sequence (0111000) and an incorrect bit sequence (0110000), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector can be expressed by the following equations (33), (34), and (35), respectively.d230=(T5−T4)2+(T5−T3)2+(T3−T1)2+(T1−T0)2=d206   (33)
                                                                                          s                  30                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        4                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        3                                                              }                                    2                                +                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              1                                                        ,                            n                                                    )                                                                    -                                              T                        1                                                              }                                    2                                +                                                      {                                                                  y                        ⁡                                                  (                                                      0                            ,                            n                                                    )                                                                    -                                              T                        0                                                              }                                    2                                -                                                                                                      ⁢                                                                    {                                                                  y                        ⁡                                                  (                                                                                    -                              3                                                        ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                      {                                                                  y                        ⁡                                                  (                                                                                    -                              2                                                        ,                            n                                                    )                                                                    -                                              T                        5                                                              }                                    2                                -                                                                                                      ⁢                                                                                          {                                                                        y                          ⁡                                                      (                                                                                          -                                1                                                            ,                              n                                                        )                                                                          -                                                  T                          3                                                                    }                                        2                                    -                                                            {                                                                        y                          ⁡                                                      (                                                          0                              ,                              n                                                        )                                                                          -                                                  T                          1                                                                    }                                        2                                                  =                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          3                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          2                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      y                    ⁡                                          (                                                                        -                          1                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        1                                            -                                              T                        0                                                              }                                    ⁢                                      y                    ⁡                                          (                                              0                        ,                        n                                            )                                                                      +                                  T                  4                  2                                +                                  T                  0                  2                                -                                  2                  ⁢                                      T                    5                    2                                                                                                          (        34        )                                                                                                      ∂                                                            s                      30                                        ⁡                                          (                      n                      )                                                                                        ∂                                      c                    ⁡                                          (                                              k                        ,                        n                                            )                                                                                  =                            ⁢                                                2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        4                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            3                                                    +                          k                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        5                                            -                                              T                        3                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            2                                                    +                          k                                                ,                        n                                            )                                                                      +                                                                                                      ⁢                                                2                  ⁢                                      {                                                                  T                        3                                            -                                              T                        1                                                              }                                    ⁢                                      u                    ⁡                                          (                                                                                                    -                            1                                                    +                          k                                                ,                        n                                            )                                                                      +                                  2                  ⁢                                      {                                                                  T                        1                                            -                                              T                        0                                                              }                                    ⁢                                      u                    ⁡                                          (                                              k                        ,                        n                                            )                                                                                  ]                                                          (        35        )            
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢            μ            ⁢                          {                                                                    s                    30                                    ⁡                                      (                    n                    )                                                  -                                  d                  30                  2                                            }                        ⁢                                          ∂                                                      s                    30                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        36        )            
12) In the case of a correct bit sequence (0110000) and an incorrect bit sequence (0111000), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                +                      2            ⁢                                                  ⁢            μ            ⁢                          {                                                -                                                            s                      30                                        ⁡                                          (                      n                      )                                                                      -                                  d                  30                  2                                            }                        ⁢                                          ∂                                                      s                    30                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        37        )            
13) In the case of a correct bit sequence (1111001) and an incorrect bit sequence (1110001), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (38), (39), and (40), respectively.d271=(T6−T5)2+(T5−T3)2+(T3−T1)2+(T2−T1)2=d247   (38)
                                          s            71                    ⁡                      (            n            )                          =                                            {                                                y                  ⁡                                      (                                                                  -                        3                                            ,                      n                                        )                                                  -                                  T                  5                                            }                        2                    +                                    {                                                y                  ⁡                                      (                                                                  -                        2                                            ,                      n                                        )                                                  -                                  T                  3                                            }                        2                    +                                    (        39        )                                                          ⁢                                            {                                                y                  ⁡                                      (                                                                  -                        1                                            ,                      n                                        )                                                  -                                  T                  1                                            }                        2                    +                                    {                                                y                  ⁡                                      (                                          0                      ,                      n                                        )                                                  -                                  T                  1                                            }                        2                    -                                                                                              ⁢                                            {                                                y                  ⁡                                      (                                                                  -                        3                                            ,                      n                                        )                                                  -                                  T                  6                                            }                        2                    -                                    {                                                y                  ⁡                                      (                                                                  -                        2                                            ,                      n                                        )                                                  -                                  T                  5                                            }                        2                    -                                                                                              ⁢                                            {                                                y                  ⁡                                      (                                                                  -                        1                                            ,                      n                                        )                                                  -                                  T                  3                                            }                        2                    -                                    {                                                y                  ⁡                                      (                                          0                      ,                      n                                        )                                                  -                                  T                  2                                            }                        2                                                                                                        ⁢                  =                                    2              ⁢                              {                                                      T                    6                                    -                                      T                    5                                                  }                            ⁢                              y                ⁡                                  (                                                            -                      3                                        ,                    n                                    )                                                      +                          2              ⁢                              {                                                      T                    5                                    -                                      T                    3                                                  }                            ⁢                              y                ⁡                                  (                                                            -                      2                                        ,                    n                                    )                                                      +                                                                                                        ⁢                              2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          y              ⁡                              (                                                      -                    1                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  1                                -                                  T                  0                                            }                        ⁢                          y              ⁡                              (                                  0                  ,                  n                                )                                              +                                                                                              ⁢                              2            ⁢                          T              1              2                                -                      T            6            2                    -                      T            2            2                                                                                                    ∂                                          s                71                            ⁡                              (                n                )                                                          ∂                          c              ⁡                              (                                  k                  ,                  n                                )                                                    =                              2            ⁢                          {                                                T                  6                                -                                  T                  5                                            }                        ⁢                          u              ⁡                              (                                                                            -                      3                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  5                                -                                  T                  3                                            }                        ⁢                          u              ⁡                              (                                                                            -                      2                                        +                    k                                    ,                  n                                )                                              +                                    (        40        )                                                          ⁢                              2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                                                            -                      1                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  2                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                  k                  ,                  n                                )                                                    ]                                        
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢                                                  ⁢            μ            ⁢                          {                                                                    s                    71                                    ⁡                                      (                    n                    )                                                  -                                  d                  71                  2                                            }                        ⁢                                          ∂                                                      s                    71                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        41        )            
14) In the case of a correct bit sequence (1110001) and an incorrect bit sequence (1111001), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                +                      2            ⁢                                                  ⁢            μ            ⁢                          {                                                -                                                            s                      71                                        ⁡                                          (                      n                      )                                                                      -                                  d                  71                  2                                            }                        ⁢                                          ∂                                                      s                    71                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        42        )            
15) In the case of a correct bit sequence (1111000) and an incorrect bit sequence (1110000), the square of Euclidean distance d2 between two paths having the same state transition, the metric difference s(n) between the equalization waveform and each of the paths having the same state transition, and the gradient vector are expressed by the following equations (43), (44), and (45), respectively.d270=(T6−T5)2+(T5−T3)2+(T3−T1)2+(T1−T0)2=d207   (43)
                                          s            70                    ⁡                      (            n            )                          =                                            {                                                y                  ⁡                                      (                                                                  -                        3                                            ,                      n                                        )                                                  -                                  T                  5                                            }                        2                    +                                    {                                                y                  ⁡                                      (                                                                  -                        2                                            ,                      n                                        )                                                  -                                  T                  3                                            }                        2                    +                                    (        44        )                                                          ⁢                                            {                                                y                  ⁡                                      (                                                                  -                        1                                            ,                      n                                        )                                                  -                                  T                  1                                            }                        2                    +                                    {                                                y                  ⁡                                      (                                          0                      ,                      n                                        )                                                  -                                  T                  0                                            }                        2                    -                                                                                              ⁢                                            {                                                y                  ⁡                                      (                                                                  -                        3                                            ,                      n                                        )                                                  -                                  T                  6                                            }                        2                    -                                    {                                                y                  ⁡                                      (                                                                  -                        2                                            ,                      n                                        )                                                  -                                  T                  5                                            }                        2                    -                                                                                              ⁢                                            {                                                y                  ⁡                                      (                                                                  -                        1                                            ,                      n                                        )                                                  -                                  T                  3                                            }                        2                    -                                    {                                                y                  ⁡                                      (                                          0                      ,                      n                                        )                                                  -                                  T                  1                                            }                        2                                                                                                        ⁢                  =                                    2              ⁢                              {                                                      T                    6                                    -                                      T                    5                                                  }                            ⁢                              y                ⁡                                  (                                                            -                      3                                        ,                    n                                    )                                                      +                          2              ⁢                              {                                                      T                    5                                    -                                      T                    3                                                  }                            ⁢                              y                ⁡                                  (                                                            -                      2                                        ,                    n                                    )                                                      +                                                                                                        ⁢                              2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          y              ⁡                              (                                                      -                    1                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  1                                -                                  T                  0                                            }                        ⁢                          y              ⁡                              (                                  0                  ,                  n                                )                                              +                      T            0            2                    -                      T            6            2                                                                                                    ∂                                          s                70                            ⁡                              (                n                )                                                          ∂                          c              ⁡                              (                                  k                  ,                  n                                )                                                    =                              2            ⁢                          {                                                T                  6                                -                                  T                  5                                            }                        ⁢                          u              ⁡                              (                                                                            -                      3                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  5                                -                                  T                  3                                            }                        ⁢                          u              ⁡                              (                                                                            -                      2                                        +                    k                                    ,                  n                                )                                              +                                    (        45        )                                                          ⁢                              2            ⁢                          {                                                T                  3                                -                                  T                  1                                            }                        ⁢                          u              ⁡                              (                                                                            -                      1                                        +                    k                                    ,                  n                                )                                              +                      2            ⁢                          {                                                T                  1                                -                                  T                  0                                            }                        ⁢                          u              ⁡                              (                                  k                  ,                  n                                )                                                    ]                                        
Consequently, the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                -                      2            ⁢                                                  ⁢                          μ              (                                                                    s                    70                                    ⁡                                      (                    n                    )                                                  -                                  d                  70                  2                                            }                        ⁢                                          ∂                                                      s                    70                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        46        )            
16) In the case of a correct bit sequence (1110000) and an incorrect bit sequence (1111000), the tap coefficient c(k,n) is updated as follows.
                              c          ⁡                      (                          k              ,                              n                +                1                                      )                          =                              c            ⁡                          (                              k                ,                n                            )                                +                      2            ⁢                                                  ⁢                          μ              (                                                -                                                            s                      70                                        ⁡                                          (                      n                      )                                                                      -                                  d                  70                  2                                            }                        ⁢                                          ∂                                                      s                    70                                    ⁡                                      (                    n                    )                                                                              ∂                                  c                  ⁡                                      (                                          k                      ,                      n                                        )                                                                                                          (        47        )            
Furthermore, the above-described tap coefficient will be updated only when a bit sequence (x00x11x) or (x11x00x) is detected.
FIG. 10 is a block diagram illustrating the construction of a conventional adaptive equalization apparatus. The conventional adaptive equalization apparatus performs adaptive equalization by using the coefficient update formulas described in 1) to 16) above.
An optical head 2 reads bit sequences which are recorded on an optical disk 1.
An analog-to-digital converter (A/D converter) 3 converts an analog playback signal, which is a recorded bit sequence read by the optical head 2, into a digital signal.
An equalizer 4 equalizes the output signal of the A/D converter 3.
A Viterbi decoder 5 subjects the equalized signal which is outputted from the equalizer 4 to maximum likelihood decoding.
A delay unit 6 carries out delay adjustment for the equalized signal which is outputted from the equalizer 4.
A delay unit 7 carries out delay adjustment for the digital playback signal which is an input to the equalizer 4.
A coefficient adaptive controller 8 adaptively controls tap coefficients which are used in the equalizer 4, on the basis of a maximum likelihood decoded bit sequence, i.e., the output signal of the Viterbi decoder 5, the output signal of the delay unit 6, and the output signal of the delay unit 7.
Hereinafter, a description will be given of the operation of the conventional adaptive equalization apparatus.
The bit sequence recorded on the optical disk 1 is read by the optical head 2 as an analog playback signal. The read analog playback signal is converted into a digital playback signal by the A/D converter 3, and equalized into PR(1,2,2,1) by the equalizer 4. The equalized digital signal is subjected to maximum likelihood decoding by the Viterbi decoder 5, thereby generating a maximum likelihood decoded bit sequence.
In order to adjust the timing with the maximum likelihood decoded bit sequence obtained by the Viterbi decoder 5, the output of the equalizer 4 is delay-adjusted by the delay unit 6, and the input to the equalizer 4 is delay-adjusted by the delay unit 7. In the coefficient adaptive controller 8, the tap coefficient is adaptively updated on the basis of the decoded bit sequence, the delay-adjusted output of the equalizer 4, and the delay-adjusted input to the equalizer 4. The updated tap coefficient is then output to the equalizer 4. In this way, the equalizer 4 adaptively equalizes the digital playback signal.
Hereinafter, an explanation will be given of the coefficient adaptive controller 8 with reference to FIG. 11. FIG. 11 is a block diagram illustrating the construction of the coefficient adaptive controller 8.
In FIG. 11, a bit sequence detection circuit 81 detects a specific bit sequence from the decoded bit sequence. A maximum likelihood decoding error calculation circuit 82 calculates 16 types of maximum likelihood decoding errors from the delay-adjusted output of the equalizer 4, and selects one out of the 16 errors by a bit sequence detection signal. A gradient vector calculation circuit 83 calculates gradient vectors of 16 types of error characteristics curved-planes from the delay-adjusted input to the equalizer 4, and selects one out of the gradient vectors by the bit sequence detection signal. An update control circuit 84 outputs a coefficient update signal on the basis of the above bit sequence detection signal. A coefficient update circuit 85 carries out updating of the tap coefficient on the basis of the coefficient update signal.
Next, a description will be given of the coefficient adaptive control method.
The bit sequence detection circuit 81 detects specific bit sequences (x00x11x) and (x11x00x) from the decoded bit sequences. Since, in the above example, there exist 16 types of specific bit sequence, the bit sequence detection circuit 81 may be constructed so as to output a corresponding number of 1 to 16 when a specific bit sequence is detected, and output a number of 0 when there is no specific bit sequence detected.
The maximum likelihood decoding error calculation circuit 82 calculates 16 types of maximum likelihood decoding errors from the delay-adjusted output of the equalizer 4, selects one out of the 16 errors on the basis of the bit sequence detection signal, and outputs the selected error to the gradient vector calculation circuit 83. The gradient vector calculation circuit 83 calculates gradient vectors of 16 types of error characteristics curved-planes on the basis of the maximum likelihood decoding error and the delay-adjusted input to the equalizer 4, selects one out of the gradient vectors according to the bit sequence detection signal, and then outputs the selected gradient vector to the coefficient update circuit 85.
The update control circuit 84 outputs a coefficient update signal according to the value of the bit sequence detection signal to the coefficient update circuit 85. The coefficient update circuit 85 adds a value, which is obtained by multiplying the gradient vector by a coefficient μ, to the tap coefficient, and outputs it as new tap coefficient. When the coefficient update signal is at LOW, the tap coefficient is not updated.
The conventional adaptive equalization apparatus shown in FIG. 10 performs coefficient adaptive control by using the maximum likelihood decoded bit sequence, which is obtained from the Viterbi decoder 5. The input and output of the equalizer 4 are delayed by the delay units 6 and 7 to make them coincide with the timing of the decoded bit sequence.
When only one survival path is left in the trellis diagram, the Viterbi decoder 5 decodes bits corresponding to the survival path. Therefore, the Viterbi decoder 5 requires a path memory having a length expected from the last survival path, and a delay equivalent to the length. For example, if the path memory has 30 stages, at least a delay equivalent to 30 clock is required.
As described above, since the delay in the Viterbi decoder 5 is considerable, the delay in the loop for adaptively controlling the tap coefficient is increased. Therefore, it takes much time to obtain the optimal tap coefficient. Further, it is difficult to follow abrupt variations in the waveform read by an optical head. Furthermore, a delay unit having a number of stages is required, resulting in an increase in circuit scale.